1 Nucleosynthesis of the Heavy Elements Three basic processes can be identified by which heavy nuclei can be built by the continuous addition of protons or neutrons: • p-process (proton) • s-process (slow neutron) • r-process (rapid neutron) Capture of protons on light nuclei tend to produce only proton-rich nu- clei. Capture of neutrons on light nuclei produce neutron-rich nuclei, but which nuclei are produced depends upon the rate at which neutrons are added. Slow capture produces nuclei near the valley of beta stability, while rapid capture (i.e., rapid compared to typical beta-decay timescales) ini- tially produces very neutron-rich radioactive nuclei that eventually beta- decay towards the valley of beta stability. Some nuclei can be built by more than one process, as illustrated in Fig. 1. Figure 1: A small portion of the chart of the nuclides, illus- trating isotopes built by the three basic processes. Only stable isotopes are shown, and letters indicate the processes that can contribute to these isotopes. 2 Neutron Capture The difference between the s-process and r-process nucleosynthesis is controlled by beta decay rates of nuclei. The s-process path lies in the valley of beta stability, but the r-process path is shifted by many units toward neutron-richness. Fig. 2 shows the beta-decay timescales of radioactive neutron-rich nuclei, the positions of stable nuclei, neutron and proton magic numbers and the r-process path. Figure 2: Chart of the nuclides illustrating beta decay timescales (color), stable isotopes (black squares), magic numbers (hori- zontal and vertical black lines), the limit of known nuclei (di- agonal jagged black line), and the r-process path (purple line). Overlaid are the relative abundances of r-process nuclei, to il- lustrate the relation between magic numbers and abundances. For small neutron velocities v, i.e., energies up to a tens of keV, neutron capture cross sections vary as σ / v−1. Thus the product σv is roughly 3 constant for thermal energies. Characteristic cross sections are about 100- 1000 millibarns (10−25 − 10−24 cm2). For temperatures around 30 keV, thermal velocities of neutrons are about 3 × 108 cm s−3. Then, the lifetime of a nucleus against neutron capture is 1 3 × 1016 − 3 × 1017 τn ∼ ∼ s: nnσv nn s-process timescales are around 104 yr, while r-process timescales are around microseconds, indicating neutron densities of 105 − 106 cm−3 and 1023 − 1024 cm−3, respectively. Neutron densities intermediate to these might be expected to yield a nucleosynthesis pattern intermediate to the s- and r- processes. However, comparison to solar-system abundances indicates that this has not occurred. Neutron capture cross sections are generally smoothly varying with A because a captured neutron has an excitation energy of about 8 MeV (µn ' µp ≈ −8 MeV) where the density of levels if very high. However, for light nuclei or magic number nuclei (N or Z 2 [2; 8; 20; 28; 50; 82; 126]) this is not the case and the neutron capture cross section are very small, comparatively, being from 1 to 10 mb. s-process A quantitative analysis can be performed with the basic assumption that, except for magic number nuclei, beta decay timescales are all very rapid compared to the neutron cature rate. The nuclear chain thus indicated is unique, because one proceeds to heavier nuclei as indicated in Fig. 2, stepping horizontally until an unstable nucleus is reached, then moving on a diagonal path increasing the proton number by 1 and decreasing the neutron number by 1. One can let NA represent the abundance of nucleus A with out reference to Z, and we can ignore abundances of radioactive nuclei. The differential equation for abundances is then _ NA = − < σv >A nn (t) NA (t) + < σv >A−1 nn (t) NA−1 (t) ; where < σv >A is thermally averaged for each nucleus A. Under the as- −1 sumption that σ / v , one can write < σv >A≈ vT σA where vT is the thermal velocity which is a constant. Defining a neutron exposure τ by dτ = vT nn (t) dt; 4 we find 0 NA = −σANA + σA−1NA−1: This equation can only be solved with a suitable boundary condition. Be- cause neutron capture cross sections on light nuclei are so small, we can effectively assume that NA(0) = 0 for A < 56 and NA(0) = N56(0) for A > 56. On the other hand, for A = 209 the chain cerminates because Bi209 is the most massive stable nucleus. Nuclei with mass 210 decay by α decay to mass 206. We will ignore this complication. 0 The above equation has the property that NA < 0 if NA > (σA−1/σA)NA−1 and vice-versa. Thus it is self-regulating: NA decreases if it is too large with respect to NA−1 and increases if it is too small. If the process operates long enough, it will come to equilibrium: NA0 = 0, in which case σANA ≈ σA−1NA−1: In-between the magic numbers, the cross sections are large compared to their differences, so locally this is a good approximation to the solution. Over a large range of A, however, this result is a poor approximation, because of the small cross sections associated with magic nuclei. A proper general soution is a little tricky, but can be analyzed by using the Laplace transform 1 ^ −st Nk (s) = Nk (t) e dt Z0 using k = A − 55. Using the fact that the Laplace transform of a derivative is 1 0 −st ^ Nk (t) e dt = sNk (s) − Nk (0) ; Z0 the transformed equations and boundary condition are: sN^1 (s) = − σ1N^1 (s) + N1 (0) ; ^ ^ ^ sNk (s) = − σkNk (s) + σk−1Nk−1 (s) : The solution of these equations, by substitution, is ^ N1 (0) σk Nk (s) = : σk s + σk Yk Here the notation Πk indicates successive multiplications of arguments from subscript k to 1. Since the product σN is expected to be smoothly varying, 5 we choose to use that product as the dependent variable. We define ^ ^ σkNk (s) −1 k (s) ≡ = (1 + s/σk) : N1 (0) Yk The solution is then the inverse Laplace transform: 1 i1 (τ) ≡ σN (τ) = ^ (s) esτ ds k k 2πi k Z−i1 k σ −σiτ k = e σi : σk − σi Xi=1 kY=6 i Figure 3: The distribution function k(τ) using σ = 100. Left: Shown as a function of k for τ = 0:5012. Right: Shown as a function of τ for k = 50. This solution encounters severe numerical difficulties at large atomic weights. The sum must be taken over about 150 terms, which differ greatly in order of magnitude but still must be included because of their near can- cellations. It is therefore necessary to find an approximate solution. To illustrate the behavior of the exact solution, consider the idealized case in which all the cross sections are equal, σk = σ. The solution in that case, starting from ^ − k (s) = (1 + s/σ) k; 6 is σ (τ) = (στ)k−1 e−στ ; k (k − 1)! which is a Poisson-like distribution in both k and τ with maxima at kmax = −1 στ + 1 and τmax = σ (k − 1). Thus, as the neutron exposure τ increases, the maximum in the distribution moves to larger values of k. The abun- dance (distribution width) at the maximum N = ∆−1 ' (2πστ)−1=2 kmax kmax decreases (increases) with k. The approximate solution begins with this result. If we can find numbers λk and mk such that s −mk s −1 1 + ' Π 1 + λ k σ k k for small s, we could write an approximate solution in the general case as (λ τ)mk−1 k −λkτ k (τ) ' λk e : Γ (mk) It can be shown that the constraint that the first three moments of k(τ) for the approximate case equal that of the general case, i.e., 1 k (τ) dτ = 1; Z0 1 k −1 τ k (τ) dτ = σi ; Z0 i X 2 1 k k 2 −1 −2 τ k (τ) dτ = σi + σi ; 0 0 1 Z Xi Xi @ A yields 2 k −1 k −1 i=1 σi σ m = ; λ = i=1 i : k k −2 k k −2 P i=1 σi Pi=1 σi This solution is valid to bPetter than 10% accuracyP, which is small compared to the numerical uncertainty in the experimental products σkNk. Each function k(τ) begins at zero, rises to a single maximum at some value τk;max and then falls exponentially to zero at large τ. Magic numbers 7 produce abrupt changes in the distributions. Often, results are expressed in terms of the number of neutrons captured per initial iron seed nucleus instead of τ, a relationship which depends upon the values of the cross sections in the capture chain. Figure 4: The distribution function A for different labels of total neutron irradiation, labelled by nc, the average number of neutrons captured per initial seed nucleus (Fe). Each s-process event results in a different abundance pattern because the exposure τ varies from event to event. An often-used approximation is that −τ/τ the probability of an event is equal to Ge o with G and τo parameters. With this approximation, the abundance patter produced overall will be 1 (λ τ)mA−1 λ τ mA σ N ' Gλ e−τ/τo A e−λAτ dτ = G A o : A A A Γ (m ) 1 + λ τ Z0 A A o That such a simple model accurately reproduces solar system s-process abundances is taken as confirmation of the general correctness of the ideas.